Theorem undif5 30540

Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)

Assertion

Ref	Expression
undif5	⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)

Proof of Theorem undif5

Step	Hyp	Ref	Expression
1		difun2 4381	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
2		disjdif2 4380	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)
3	1, 2	syl5eq 2783	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1543   ∖ cdif 3850   ∪ cun 3851   ∩ cin 3852  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-nul 4224

This theorem is referenced by:  fressupp  30696